Zu Chongzhi 祖沖之

Zu Chongzhi, courtesy name Wenyuan, was a prominent Chinese mathematician and astronomer during the Liu-Song and Southern Qi Dynasties, best known for calculating pi to seven decimal points.

Complete Dictionary of Scientific Biography

TSU CH’UNG-CHIH (b, Fan-yang prefecture [modern Hopeh province], China, ca.a.d. 429; d, China, ca. a.d. 500) mathematics.

Tsu Ch’ung-chih was in the service of the emperor Hsiao-wu (r. 454–464) of the Liu Sung dynasty, first as an officer subordinate to the prefect of Nan-hsü (in modern Kiangsu province), then as an officer on the military staff in the capital city of Chien-k’ang (modern Nanking). During this time he also carried out work in mathematics and astronomy; upon the death of the emperor in 464, he left the imperial service to devote himself entirely to science. His son, Tsu Keng, was also an accomplished mathematician.

Tsu Ch’ung-chih would have known the standard works of Chinese mathematics, the Chou-pi suan-ching (“Mathematical Book on the Measurement With the Pole”), the Hai-tao suan-ching (“Sea-island Manual”),(“Mathematical Manual in Nine Chapters”), of which Liu Hui had published a new edition, with commentary, in 263. Like his predecessors, Tsu Ch’ung-chih was particularly interested in determining the value of π. This value was given as 3 in the Chou-pi suan-ching; as 3.1547 by Liu Hsin (d.23); as or , by Chang Heng (78-139); and as , that is 3.1547 by Wan Fan (219-257).Since the original works of these mathematicians have been lost, it is impossible to determine how these values were obtained, and the earliest extant account of the process is that given by Liu Hui, who reached an approximate value of 3.14. Late in the fourth century, Ho Chēng-tein arrived at an approximate value of , or 3. 1428.

Tsu Ch’ung-chih’s work toward obtaining a more accurate value for π is chronicled in the calendrical chapters (Lu-li chih) of the Sui-shu, an official history of the Sui dynasty that was compiled in the seventh century by Wei Cheng and others. According to this work.

Tsu ch’ung-chih further devised a precise method. Taking a circle of diameter 100,000,000, which he considered to be equal to one chang [ten ch’ih, or Chinese feet, usually slightly greater than English feet], he found the circumference of this circle to be less than 31,415,927 chang, but greater than 31,415,926 chang,[He deduced from these results] that the accurate value of the circumference must lie between these two values. Therefore the precise value of the ratio of the circumference must lie between theses two values. Therefore the precise value of the ratio of the circumference of a circle to its diameter is a 355 to 113, and the approximate value is as 22 to 7.

The Sui-shu historians then mention that Tsu Ch’ung-chih’s work was lost, probably because his methods were so advanced as to be beyond the reach of other mathematicians, and for this reason were not studied or preserved. In his Chun-suan shih Lung’ung (“Collected Essays on the History of Chinese Mathematics” [1933]), Li Yen attempted to establish the method by which Tsu Ch’ung-chih determined that the accurate value of π lay between 3.1415926 and 3.1415927, or .

It was his conjecture that

“As , Tsu Ch’ung-chih must have set forth that, by the equality

one can deduce that

x=15.996y, that is that x=16y.

Therefore

For the derivation of

When a, b, c, and d are positive integers, it is easy to confirm that the inequalities

hold, If these inequalities are taken into consideration, the inequalities

may be derived.

Ch’ien Pao-tsung, in Chung-kuo shu-hsüeh-shih (“History of Chinese Mathematics“[1964]), assumed that Tsu Ch’ung-chih used the inequality

S2n < S < S2n + (S2n – Sn),

Where S2n is the perimeter of a regular polygon of 2n sides inscribed within a circle of circumfernce S, while Sn is the perimeter of a regular polygon of n sides inscribed within the same circle. Ch’ien Pao-tsung thus found that

S12288 = 3.14159251

and

S24576 = 3.14159261

resulting in the inequality

3.10415926< π < 3.1415927.

Of Tsu Ch’ung-chih’s astronomical work, the most important was his attempt to reform the calendar. The Chinese calendar had been based upon a cycle of 235 lunations in nineteen years, but in 462 Tsu Ch’ung-chih suggested a new system, the Ta-ming calendar, based upon a cycle of 4,836 lunations in 391 years. His new calendar also incorporated a value of forty-five years and eleven months a tu (365/4 tu representing 360°) for the precession of the equinoxes. Although Tsu Ch’ung-chih’s powerful opponent Tai Fa-hsing strongly denounced the new system, the emperor Hsiao-Wu intended to adopt it in the year 464, but he died before his order was put into effect. Since his successor was strongly influenced by Tai Fahsing, the Ta-ming calendar was never put into official use.

BIBLIOGRAPHY

On Tsu Ch’ung-chilh and his works see Li Yen, Chung-suan-shih lun-ts’ung (“Collected Essays on the History of Chinese Mathematics”). I–III (Shanghai 1933–1934), IV (Shanghai, 1947), I–V (Peking, 1954–1955); Chung-kuo shu-hsüeh ta-kang (“Outline of Chinese Mathematics” Shanghai 1931, repr. Peking 1958), 45–50; Chung-kuo suan-hsüeh-shih (“History of Chinese Mathematics” Shanghai, 1937, repr. Peking, 1955); “Tsu Ch’ung-chih, Great Mathematician of Ancient China,” in People’s China (1956), 24; and Chung-kuo ku-tai shu-hsüeh shih-hua (“Historical Description of the Ancient Mathematics of China” Peking, 1961), written with Tu Shih-jan.

See also Ch’ien Pao-tsung, Chung-kuo shu-hsüeh-shih (“History of Chinese Mathematics” Peking, 1964), 83–90; Chou Ch’ing-shu, “Wo-kuo Ku-tai wei-ta ti k’o-hsüeh-chia; Tsu Ch’ung-chih” (“A Great Scientist of Ancient China; Tsu Ch’ung-chih”), in Li Kuang-pi and Ch’ien Chün-hua, Chung-kuo K’o-hsüeh chi-shu fa-ming ho k’o-hsü chi-shu jēn-wu lun-chi (“Essays on Chinese Discoveries and Inventions in Science and Technology and the Men who Made Them” Peking, 1955), 270–282; Li Ti, Ta k’o-hsüeh-chia Tsu Ch’ung-chih (“Tsu Ch’ung-chih the Great Scientist” Shanghai, 1959); Ulrich Libbrecht, Chinese Mathematics in the Thirteenth Century (Cambridge, Mass., 1973), 275–276; Mao I-shēng, “Chung-kuo Yüan-chou-lü lüeh-shih” (“Outline History of π in China”), in K’o-hsüeh, 3 (1917), 411; Mikami Yashio, Development of Mathematics in China and Japan (Leipzig, 1912), 51; Joseph Needham, Science and Civilization in China, III (Cambridge, 1959), 102; A.P. Youschkevitch, Geschichte der Mathematik im Mittelalter (Leipzig, 1964), 59; and Yen Tun-chieh, “Tsu Keng Pieh chuan” (“Special Biography of Tsu Keng”) in K’o-hsüeh 25 (1941), 460.

Akira Kobori

《齊書》卷52

祖沖之字文遠，范陽薊人也。祖昌，宋大匠卿。父朔之，奉朝請。

沖之少稽古，有機思。宋孝武使直華林學省，賜宅宇車服。解褐南徐州迎從事，公府參軍。

宋元嘉中，用何承天所制曆，比古十一家爲密，沖之以爲尚疏，乃更造新法。上表曰：

臣博訪前墳，遠稽昔典，五帝次，三王交分，《春秋》朔氣，《紀年》薄蝕，談、遷載述，彪、固列志，魏世注歷，晉代《起居》，探異今古，觀要華戎。書契以降，二千餘稔，日月離會之徵，星度疏密之驗。專功耽思，咸可得而言也。加以親量圭尺，躬察儀漏，目盡毫氂，心窮籌筴，考課推移，又曲備其詳矣。
然而古曆疏舛，類不精密，羣氏糾紛，莫審其會。尋何承天所上，意存改革，而置法簡略，今已乖遠。以臣校之，三睹厥謬，日月所在，差覺三度，二至晷景，幾失一日，五星見伏，至差四旬，留逆進退，或移兩宿。分至失實，則節閏非正；宿度違天，則伺察無准。臣生屬聖辰，詢逮在運，敢率愚瞽，更創新曆。
謹立改易之意有二，設法之情有三。改易者一：以舊法一章，十九歲有七閏，閏數爲多，經二百年輙差一日。節閏旣移，則應改法，歷紀屢遷，寔由此條。今改章法三百九十一年有一百四十四閏，令却合周、漢，則將來永用，無復差動。其二：以堯典云「日短星昴，以正仲冬」。以此推之，唐世冬至日，在今宿之左五十許度。漢代之初，卽用秦曆，冬至日在牽牛六度。漢武改立《太初曆》，冬至日在牛初。後漢四分法，冬至日在斗二十二。晉世姜岌以月蝕檢日，知冬至在斗十七。今參以中星，課以蝕望，冬至之日，在斗十一。通而計之，未盈百載，所差二度。舊法竝令冬至日有定處，天數旣差，則七曜宿度，漸與舛訛。乖謬旣著，輙應改易。僅合一時，莫能通遠。遷革不已，又由此條。今令冬至所在歲歲微差，却檢漢注，竝皆審密，將來久用，無煩屢改。又設法者，其一：以子爲辰首，位在正北，爻應初九升氣之端，虛爲北方列宿之中。元氣肇初，宜在此次。前儒虞喜，備論其義。今曆上元日度，發自虛一。其二：以日辰之號，甲子爲先，歷法設元，應在此歲。而黃帝以來，世代所用，凡十一曆，上元之歲，莫值此名。今曆上元歲在甲子。其三：以上元之歲，歷中衆條，竝應以此爲始。而《景初曆》交會遲疾，元首有差。又承天法，日月五星，各自有元，交會遲疾，亦竝置差，裁得朔氣合而已，條序紛錯，不及古意。今設法日月五緯交會遲疾，悉以上元歲首爲始，羣流共源，庶無乖誤。
若夫測以定形，據以實效。懸象著明，尺表之驗可推；動氣幽微，寸管之候不忒。今臣所立，易以取信。但綜覈始終，大存緩密，革新變舊，有約有繁。用約之條，理不自懼，用繁之意，顧非謬然。何者？夫紀閏參差，數各有分，分之爲體，非不細密，臣是用深惜毫釐，以全求妙之准，不辭積累，以成永定之制，非爲思而莫知，悟而弗改也。若所上萬一可採，伏願頒宣羣司，賜垂詳究。

事奏。孝武令朝士善曆者難之，不能屈。會帝崩，不施行。出爲婁縣令，謁者僕射。

初，宋武平關中，得姚興指南車，有外形而無機巧，每行，使人於內轉之。昇明中，太祖輔政，使沖之追修古法。沖之改造銅機，圓轉不窮，而司方如一，馬均以來未有也。時有北人索馭驎者，亦云能造指南車，太祖使與沖之各造，使於樂遊苑對共校試，而頗有差僻，乃毀焚之。永明中，竟陵王子良好古，沖之造欹器獻之。

文惠太子在東宮，見沖之曆法，啓世祖施行，文惠尋薨，事又寢。轉長水校尉，領本職。沖之造《安邊論》，欲開屯田，廣農殖。建武中，明帝使沖之巡行四方，興造大業，可以利百姓者，會連有軍事，事竟不行。

沖之解鍾律，博塞當時獨絕，莫能對者。以諸葛亮有木牛流馬，乃造一器，不因風水，施機自運，不勞人力。又造千里船，於新亭江試之，日行百餘里。於樂遊苑造水碓磨，世祖親自臨視。又特善。永元二年，沖之卒。年七十二。著《易老莊義釋》、《論語孝經注》，《九章》造《綴述》數十篇。